A quality control inspector evaluates a batch of 16 precision components. Exactly 10 of th… — GRE General Test

A quality control inspector evaluates a batch of $16$ precision components. Exactly $10$ of the components meet all engineering specifications, while $6$ have minor surface defects. If the inspector randomly selects $2$ components from the batch one after another without replacement, what is the probability that both selected components meet all engineering specifications?

Cevap: 0.375

Cevap

The probability that both selected components meet all engineering specifications is

0.375

(or

\frac{3}{8}

Since the components are selected without replacement, the outcome of the second draw depends on the outcome of the first draw. The probability of selecting a qualifying component first is

\frac{10}{16}

. Given that a qualifying component was drawn first,

9

qualifying components remain out of

15

total components. The joint probability of both events occurring is

\frac{10}{16} \times \frac{9}{15} = \frac{90}{240} = \frac{3}{8} = 0.375

Adım Adım Çözüm

Determine the probability of selecting a component meeting specifications on the first draw.

P(E_1) = \frac{10}{16} = \frac{5}{8}

There are

10

qualifying components out of

16

total components.

Determine the conditional probability of selecting a second component meeting specifications, given that the first component selected also met specifications.

P(E_2 \mid E_1) = \frac{9}{15} = \frac{3}{5}

Because sampling is done without replacement,

9

qualifying components remain out of

15

total remaining components.

Apply the multiplication rule for dependent events to calculate the probability of both events occurring.

P(E_1 \cap E_2) = P(E_1) \times P(E_2 \mid E_1) = \frac{5}{8} \times \frac{3}{5} = \frac{3}{8} = 0.375

For dependent events, the joint probability is the product of the first event's probability and the conditional probability of the second event.

Anahtar Kavram

Probability of Dependent Events without Replacement

Tahmini Süre:1m 15s

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